A120351 Even numbers k such that the number of odd divisors r and the number of even divisors s are both divisors of k.

2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 72, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206

Offset

1,1

Comments

Since s=0 if k is odd, the number k is necessarily even and then s is always a multiple of r. Note that t=r+s may not be a divisor even if both r and s are divisors. For example, if k=144, then r=3, s=12, but t=r+s=15.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = n is even, r = number of odd divisors of n, s = number of even divisors of n, are all divisors of n.

Example

16 is a term since r=1 and s=4 are both divisors.

Maple

with(numtheory); A:=[]: N:=10^4/2: for w to 1 do for k from 2 to N do n:=2*k; S:=divisors(n); r:=nops( select(z->type(z, odd), S) ); s:=nops( select(z->type(z, even), S) ); if andmap(z -> n mod z = 0, [r, s]) then A:=[op(A), n]; print(n, r, s); fi; od od; A;

Mathematica

aQ[n_] := Divisible[n, (ev = DivisorSigma[0, n/2])] && Divisible[n, DivisorSigma[0, n] - ev]; Select[Range[2, 206, 2], aQ] (* Amiram Eldar, Nov 02 2019 *)

Crossrefs

Cf. A033950, A049439, A057265.

Sequence in context: A090127 A057910 A336066 * A022305 A103799 A175817

Adjacent sequences: A120348 A120349 A120350 * A120352 A120353 A120354

Keyword

nonn

Author

Walter Kehowski, Jun 24 2006

Extensions

Term 2 inserted by Amiram Eldar, Nov 02 2019

Status

approved